# Fractal–Fractional Michaelis–Menten Enzymatic Reaction Model via Different Kernels

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Schemes of Fractal–Fractional Michaelis–Menten Enzymatic Reaction Model

#### 2.1. Preliminaries and Notation

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

#### 2.2. FFMMER Scheme via the Power-Law Kernel

#### 2.3. FFMMER Scheme via the Exponential Decay Kernel

#### 2.4. FFMMER Scheme via the Mittag-Leffler Kernel

## 3. Numerical Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Comparison between the numerical solutions of Equations (36)–(39) via power-law kernel and numerical solution based on finite differences method for $\varrho =1,k=1,\delta =1,\gamma =2,\sigma =1$, and $h=0.01.$ Black solid line: numerical solutions of Equations (36)–(39); red dashed line: numerical solutions of Equations (12)–(15) using finite differences method. (

**a**) ${\alpha}_{1}\left(t\right),$ (

**b**) ${\beta}_{1}\left(t\right),$ (

**c**) ${\alpha}_{2}\left(t\right),$ and (

**d**) ${\beta}_{2}\left(t\right)$.

**Figure 2.**Absolute error between the numerical solutions of Equations (82)–(85) and [24] based on Mittag-Leffler kernel for $\varrho =0.9,k=0.8,\delta =1,\gamma =2,\sigma =1$, and $h=0.0003.$ (

**a**) Absolute error for ${\alpha}_{1}\left(t\right),$ (

**b**) Absolute error for ${\beta}_{1}\left(t\right),$ (

**c**) Absolute error for ${\alpha}_{2}\left(t\right),$ and (

**d**) Absolute error for ${\beta}_{2}\left(t\right)$.

**Figure 3.**The numerical solutions of the fractal–fractional Michaelis–Menten enzymatic reaction for $\varrho =1,k=0.8,\delta =1,\gamma =2,\sigma =1$, and $h=0.003.$ (

**a**) The numerical solutions of Equations (36)–(39) based on power-law kernel; (

**b**) The numerical solutions of Equations (62)–(65) based on exponential decay kernel; (

**c**) The numerical solutions of Equations (82)–(85) based on Mittag-Leffler kernel (orange line: ${\alpha}_{1}\left(t\right)$; red line: ${\beta}_{1}\left(t\right)$; green color: ${\alpha}_{2}\left(t\right)$; blue line: ${\beta}_{2}\left(t\right))$.

**Figure 4.**The numerical solutions of the fractal–fractional Michaelis-Menten enzymatic reaction for $\varrho =0.8,k=0.9,\delta =1,\gamma =2,\sigma =1,$ and $h=0.003.$ (

**a**) The numerical solutions of Equations (36)–(39) based on power-law kernel; (

**b**) The numerical solutions of Equations (62)–(65) based on exponential decay kernel; (

**c**) The numerical solutions of Equations (82)–(85) based on Mittag-Leffler kernel (orange line: ${\alpha}_{1}\left(t\right)$; red line: ${\beta}_{1}\left(t\right)$; green line: ${\alpha}_{2}\left(t\right)$; blue line: ${\beta}_{2}\left(t\right))$.

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**MDPI and ACS Style**

Alqhtani, M.; Saad, K.M.
Fractal–Fractional Michaelis–Menten Enzymatic Reaction Model via Different Kernels. *Fractal Fract.* **2022**, *6*, 13.
https://doi.org/10.3390/fractalfract6010013

**AMA Style**

Alqhtani M, Saad KM.
Fractal–Fractional Michaelis–Menten Enzymatic Reaction Model via Different Kernels. *Fractal and Fractional*. 2022; 6(1):13.
https://doi.org/10.3390/fractalfract6010013

**Chicago/Turabian Style**

Alqhtani, Manal, and Khaled M. Saad.
2022. "Fractal–Fractional Michaelis–Menten Enzymatic Reaction Model via Different Kernels" *Fractal and Fractional* 6, no. 1: 13.
https://doi.org/10.3390/fractalfract6010013